Chapter 2

Theory on Mechanics of Solder Materials

Abstract Chapter 2 reviews the fundamental theory on mechanics of solder materials. As solder materials are subject to high operating temperatures relative to their melting point, the thermo-mechanical deformation response of the solder is dependent on both temperature and strain-rate conditions. Hence, the theory on mechanics of solder materials will focus on elastic-plastic-creep and viscoplastic models for describing the thermo-mechanical deformation response of lead-free solder materials operating over a wide range of temperatures (À40 C to +125 C) and strain rates (0.0001–1,000 sÀ1).

Solder materials in electronic packaging assemblies are subjected to thermomechanical loads during accelerated reliability tests and in service operation.
Accelerated Thermal Cycling (ATC) tests subject the solder joints to extreme temperatures; for example from À40 C to +125 C. Such cyclic thermo-mechanical induced deformations in the package assembly cause the solder material to develop severe cyclic inelastic strains and cumulative fatigue damage resulting in failure of the solder joints [1, 2].
The stress, strain, and strain energy density components in solder materials can be computed numerically, if the governing material constitutive model has been developed from mechanics of materials tests. Materials testing and characterization of the elastic and inelastic deformation behavior are needed to derive the constitutive models to describe the solder deformation behavior in an elasticplastic-creep model or a viscoplastic model approach [3].
Thermo-mechanical fatigue analysis in solder joints of electronic assemblies will require a stress and strain analysis approach and this often involves 2D and 3D modeling of the package assembly subject to the design or reliability test condition
[4–6]. Fatigue damage driving force parameters such as stress-range, strain-range, and inelastic strain energy density per cycle can be computed and used in fatigue life prediction models.

J.H.L. Pang, Lead Free Solder: Mechanics and Reliability,
DOI 10.1007/978-1-4614-0463-7_2, # Springer Science+Business Media, LLC 2012

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2.1

2 Theory on Mechanics of Solder Materials

Thermo-Mechanical Stress and Strain Analysis

An electronic package assembly is made up of different deformable materials joined by solder materials that undergo extensive amounts of deformations when subject to cyclic thermal loading. It is essential to understand the basic governing equations of thermo-mechanical strain in solder joints induced by mismatch in thermo-mechanical deformations [1, 2]. For a simple case, it can be assumed that the package is stress-free at a uniform temperature where there are no external or residual forces present. It is also assumed that the velocity and displacement of every element in the package at the instantaneous reference state is assumed to be small and the fundamental strain–displacement relation is given by,


1 @ui @uj eij ¼ þ :
2 @xj @xi

(2.1)

Hence, the governing equations for the isotropic thermal stresses for electronic packaging structure with steady-state heat flow condition (∂T/∂t ¼ 0) are given by, dij  
@ @T
¼0
@xi @xj

(2.2)

sij ¼ lekk dij þ 2Geij À bdij ðT À T0 Þ

(2.3)

with the multi-axial strain components in a solid element given by, ex ¼ gxy ¼

@u
;
@x

@u @v þ ;
@y @x

ey ¼

gyz ¼

@v
;
@y

ez ¼

@v @w þ ;
@z @y

@w
@z

gxz ¼

(2.4)
@w @u þ @x @z

(2.5)

In three-dimensional analysis, a solid is subjected to multi-axial states of stresses and strains. The principal stresses and strains in the three principal planes,
1, 2, and 3 can be computed from a mechanics of materials analysis or by a numerical modeling approach employing a finite element analysis (FEA) method. s1 >s2 >s3 and e1 >e2 >e3 ; where s1, s2, and s3 are the principal stresses, and